H2 Maths 5.2 - Maclaurin Series: Multiplying Two Expansions

Step 1 - Set up the problem

We want the Maclaurin series of e to the two x times cosine x, up to and including the x cubed term.

Step 1 - Set up the problem

The strategy is to expand each factor separately using standard templates,

Step 1 - Set up the problem

then multiply the truncated polynomials and collect by power of x.

Step 2 - Expand e to the 2x

Start from the standard e to the x series.

Step 2 - Expand e to the 2x

Substitute two x in place of x throughout.

Step 2 - Expand e to the 2x

Two x all squared over two factorial is four x squared over two, which simplifies to two x squared.

Step 2 - Expand e to the 2x

Two x all cubed over three factorial is eight x cubed over six, which simplifies to four thirds x cubed.

Step 3 - Write the cos x terms needed

Cosine x has only even powers: one minus x squared over two factorial plus x to the four over four factorial, and so on.

Step 3 - Write the cos x terms needed

For our product up to x cubed, we only need cosine x up to x squared.

Step 3 - Write the cos x terms needed

The x to the four term, multiplied by the constant one from e to the two x, would give x to the four - which we discard.

Step 3 - Write the cos x terms needed

So we truncate cosine x at x squared.

Step 4 - Multiply and collect by power

Now multiply the two truncated series term by term, keeping only powers up to x cubed.

Step 4 - Multiply and collect by power

The constant and x and x squared rows each come from one cross-product.

Step 4 - Multiply and collect by power

The x cubed row gets two contributions: four thirds x cubed from the e to the two x factor,

Step 4 - Multiply and collect by power

and two x times negative x squared over two, which is negative x cubed.

Step 4 - Multiply and collect by power

Four thirds minus one is one third.

Step 5 - Final answer and exam watch points

So the Maclaurin expansion of e to the two x cosine x up to x cubed is: one plus two x plus three halves x squared plus one third x cubed.

Step 5 - Final answer and exam watch points

Three exam watch points.

Step 5 - Final answer and exam watch points

Keep factorial denominators exact - do not evaluate two factorial to two until the final simplification.

Step 5 - Final answer and exam watch points

State the remainder order explicitly: plus order x to the four.

Step 5 - Final answer and exam watch points

Both e to the x and cosine x converge for all x, so no validity restriction is needed here.